What probability means
Probability as a number from 0 to 1, the words impossible, unlikely, even, likely and certain, and the chance of a coin or a die.
Probability is a number that tells you how likely something is to happen. You meet it every day: a 70% chance of rain, a coin flip at the start of a football match, a board game with a die. Many people say that the maths of probability began in 1654. That year two French thinkers, Blaise Pascal and Pierre de Fermat, wrote letters to each other about questions from games of chance, such as how to share the money fairly when a game is stopped before the end. Today the same ideas are used in weather forecasts, medicine and insurance. The good news is that the basic idea is simple, and you can learn it with a coin and a die.
Outcomes and events
When you flip a coin or roll a die, each possible result is called an outcome. A die has 6 outcomes: 1, 2, 3, 4, 5 and 6. A coin has 2 outcomes: heads and tails. An event is the thing you are asking about, for example "the die shows an even number". An event can be made of one outcome or of several.
A fair coin and a fair die have outcomes that are equally likely: each one has the same chance. Then the probability of an event is easy to find. Count the outcomes that make the event happen, and divide by the number of all the outcomes.
Remember
Probability = (outcomes that work) ÷ (all outcomes)
This works when every outcome is equally likely. A die with 3 faces that work out of 6 gives 3/6.
The probability scale from 0 to 1
A probability is always a number from 0 to 1. It can never be less than 0, and it can never be more than 1. You can write it as a fraction, a decimal or a percent: 1/2, 0.5 and 50% all mean the same thing. Each part of the scale has a word.
| Probability | Word | Example |
|---|---|---|
| 0 | Impossible: it can never happen | Rolling a 7 on a die |
| Close to 0, like 1/10 | Unlikely | Rolling a 6 on a die (1/6) |
| 1/2 | Even chance: as likely as not | Heads on a coin |
| Close to 1, like 4/5 | Likely | Rolling more than 1 on a die (5/6) |
| 1 | Certain: it always happens | Rolling a number from 1 to 6 |
Impossible and certain
The two ends of the scale are the easiest questions, but they catch people who calculate before they think.
Example 1
You roll a die. What is the probability of getting an 8?
- A die has the numbers 1, 2, 3, 4, 5 and 6.
- No face shows an 8, so 0 outcomes work.
- 0/6 = 0.
Answer: 0. It is impossible.
Example 2
A bag has 4 green marbles and nothing else. You take one without looking. What is the probability that it is green?
- There are 4 marbles, and all 4 are green.
- 4/4 = 1.
Answer: 1. It is certain.
A coin
A coin has two sides, and both are equally likely. Heads is 1 of the 2 outcomes, so the probability of heads is 1/2. The probability of tails is also 1/2. Heads and tails are equally likely, so the answer to "which is more likely, heads or tails?" is "both are equally likely".
A die
A normal die is a cube with 6 faces, numbered 1 to 6. For any question about one roll, the bottom of the fraction is 6. Write down the faces that work, count them, and put that number on top.
Example 3
You roll a die. What is the probability of getting a number less than 4?
- Faces that work: 1, 2 and 3. (4 is not less than 4.)
- That is 3 of the 6 faces: 3/6.
- Simplify: divide the top and the bottom by 3. 3/6 = 1/2.
Answer: 1/2
To simplify a fraction, divide the top and the bottom by the same number until you cannot go further. So 2/6 = 1/3 and 4/6 = 2/3. Answers are usually written in this simplest form.
Some questions use special kinds of numbers. Here is what works on a die:
| The event | Faces that work | Probability |
|---|---|---|
| An even number | 2, 4, 6 | 3/6 = 1/2 |
| A prime number (a number with exactly two factors, 1 and itself) | 2, 3, 5 | 3/6 = 1/2 |
| A square number (a number times itself) | 1, 4 | 2/6 = 1/3 |
| A factor of 6 (it divides 6 exactly) | 1, 2, 3, 6 | 4/6 = 2/3 |
| More than 4 | 5, 6 | 2/6 = 1/3 |
Which is more likely?
To compare two events, find the probability of each and see which number is bigger. With a bag of marbles or a spinner, the color with the most marbles or the most parts is the most likely.
Example 4
This spinner has 6 equal parts. Which color is the arrow least likely to stop on?
- Count the parts: green 3, red 2, blue 1.
- Blue has the fewest parts: 1/6.
Answer: Blue
Example 5
Which is more likely: rolling a 5 on a die, or getting tails with a coin?
- A 5 on a die: 1/6.
- Tails: 1/2 = 3/6.
- 3/6 is bigger than 1/6.
Answer: Tails on the coin
Some comparisons are harder, and they fool many people:
- HHHHH or HTHTT? (H = heads, T = tails.) Five heads in a row looks special, but every exact list of 5 flips has the same probability, 1/2 × 1/2 × 1/2 × 1/2 × 1/2 = 1/32. They are equally likely.
- With two coins, exactly one head or two heads? Exactly one head happens in 2 ways (HT and TH), two heads in only 1 way (HH). One head is more likely.
- Two dice: which total is most likely? 7, because it can be made in 6 ways. The lesson on counting outcomes shows all 36 results.
- At least one 6 in two rolls, or heads in one flip? At least one 6 is 11/36, which is less than 18/36 = 1/2. Heads is more likely.
Try it yourself
1.An event has probability 9/10. Which word describes it best: impossible, unlikely, even chance, likely or certain?
Show the answer
Likely
9/10 is close to 1, but it is not 1, so it is likely and not certain.
2.A bag has 6 red marbles. You take one without looking. What is the probability that it is yellow?
Show the answer
0
There are no yellow marbles in the bag, so yellow is impossible.
3.You roll a die. What is the probability of getting an odd number?
Show the answer
1/2
The odd faces are 1, 3 and 5. That is 3 of the 6 faces: 3/6 = 1/2.
4.You roll a die. What is the probability of getting a number greater than 2?
Show the answer
2/3
The faces 3, 4, 5 and 6 work. That is 4 of the 6 faces: 4/6 = 2/3.
5.You roll a die. What is the probability of getting a factor of 4?
Show the answer
1/2
The factors of 4 are 1, 2 and 4. That is 3 of the 6 faces: 3/6 = 1/2.
6.Which is more likely: rolling a number less than 3 on a die, or getting heads with a coin?
Show the answer
Heads on the coin
Less than 3 means 1 or 2: 2/6 = 1/3. Heads is 1/2, which is bigger.
Ready to test yourself?
Play Probability: 20 seconds per question, and it gets harder as you go.